Search

It is shown that there exist arbitrarily large natural numbers $$N$$ and distinct nonnegative integers $$n_1,\dots,n_N$$ for which the number of zeros on $$[-\pi,\pi)$$ of the trigonometric polynomial $$\sum_{j=1}^N \cos(n_j t)$$ is $$O(N^{2/3}\log^{2/3} N)$$ .

Suppose that N = 2n and N1 = 2n-1, where n is a natural number. Denote by ℂN the space of complex N-periodic sequences with standard inner product. For any N-dimensional complex nonzero vector (b0, b1,..., bN-1) satisfying the condition $${\left| {{b_l}} \right|^2} + {\left| {{b_{l + {N_1}}}} \right|^2} \leq \frac{2}{{{N^2}}},\;\;\;l = 0,1,...,{N_1} - 1,$$ we find sequences u0, u1,...., ur ∈ ℂN such that the system of their binary shifts is a Parseval frame for ℂN. It is noted that the vector (b0, b1,..., bN-1) specifies the discrete Walsh transform of the sequence u0, and the choice of this vector makes it possible to adapt the proposed construction to the signal being processed according to the entropy, mean-square, or some other criterion.

We solve a nonlinear additive problem in prime and natural numbers whose binary expansion is of a special form.

The author attempts to change and supplement the standard scheme of partitions of integers in number theory to make it completely concur with the Bohr–Kalckar correspondence principle. In order to make the analogy between the the atomic nucleus and the theory of partitions of natural numbers more complete, to the notion of defect of mass author assigns the “defect” $$\overline {\left\{ a \right\}} $$ = [a + 1] − a of any real number a (i.e., the fractional value that must be added to a in order to obtain the nearest larger integer). This allows to carry over the Einstein relation between mass and energy to a relation between the whole numbers M and N, where N is the number of summands in the partition ofM into positive summands, as well as to define the forbidding factor for the number M, and apply this to the Bohr–Kalckar model of heavy atomic nuclei and to the calculation of the maximal number of nucleons in the nucleus.

Let N0 be the set of natural numbers whose binary expansions have an even number of 1’s, and let N1 = N\N0. In this paper, we obtain asymptotic formulas for the number of primes p not exceeding X and such that p ∈ N i , p + 1 ∈ N j , where i and j take values 0 and 1 independently of each other.

An asymptotic formula in the generalized Estermann ternary problem for noninteger powers with almost equal summands dealing with the representation of a sufficiently large natural number as the sum of two primes and the integer part of a noninteger power of a natural number is proved.

Let $\mathfrak{S}_n $ be the semigroup of mappings of a set of n elements into itself, let A be a fixed subset of the set of natural numbers ℕ, and let V n (A) be the set of mappings from $\mathfrak{S}_n $ for which the sizes of the contours belong to the set A. Mappings from it V n (A) are usually called A-mappings. Consider a random mapping σ n uniformly distributed on V n (A). It is assumed that the set A possesses asymptotic density ϱ, including the case ϱ = 0. Let ξ in be the number of connected components of a random mapping σ n of size i ∈ ℕ. For a fixed integer b ∈ ℕ, as n→∞, the asymptotic behavior of the joint distribution of random variables ξ1n , ξ2n ,..., ξ bn is studied. It is shown that this distribution weakly converges to the joint distribution of independent Poisson random variables η 1, η 2,..., η b with some parameters λ i = Eη i , i ∈ ℕ.

We establish a nontrivial estimate for a short trigonometric sum of the form Ωx − y < n ≤ xe(α[nc]), where u ≥ √2cx ℒ6A, A ≥ 1 is a fixed number, ℒ = ln x and c is a noninteger satisfying the conditions $$1 < c \leqslant \log _2 L - \log _2 \ln L^{6A} , \left\| c \right\| \geqslant \left( {2^{\left[ c \right] + 1} - 1} \right)\left( {A + 1} \right)L^{ - 1} \ln L.$$

We prove an asymptotic formula for the number of representations of a sufficiently large natural number N as the sum of two primes p1 and p2 and the cube of a natural numbermsatisfying the conditions |pi − N/3| ≤ H, |m3 − N/3| ≤ H, H ≥ N5/6ℒ10.

For any ɛ > 0 and all primes p, with the exception of primes from a set with relative zero density, there exists a natural number a ≤ (log p)3/2+ɛ for which the congruence a p−1 ≡ 1 (modp 2) does not hold.

We obtain an asymptotic formula for the mean value of the function τk(n), which is the number of solutions of the equation x1 … xk = n in natural numbers x1, …, xk, in some special sequences of natural numbers.

We construct an example of a double sequence a of nonnegative numbers that are monotone decreasing to zero in the first index for any fixed value of the second index and two Hadamard lacunary sequences of natural numbers such that the double trigonometric lacunary monotone series with the coefficients a constructed from the first lacunary sequence is squaredivergent almost everywhere and the one constructed from the second lacunary sequence is squareconvergent almost everywhere.

For the approximation of functions, a uniform grid of values of the arguments is often chosen. In this connection, it is natural to pose the question of how the discrete norm on a given grid relates to the uniform norm of the corresponding function on a given set. In the comparatively recent paper [1], SheilSmall showed that, for algebraic polynomials P of degree n and natural numbers N > n, the following estimate holds

We prove that almost all natural numbers n satisfying the congruence n ≡ 3 (mod 24), n ≢ 0 (mod 5), can be expressed as the sum of three squares of primes, at least one of which can be written as 1 + x2 + y2.

We present recurrence formulas for the number of partitions of a natural number n whose parts must be not less than m. A simple proof of Euler’s formula for the number of partitions is given. We construct the triangle of partitions, put forward conjectures concerning the properties of the triangle, and propose an algorithm for calculating the partitions. An original graphical interpretation for the partition function is presented.

For a gap sequence of natural numbers {nk}k=1∞, for a nondecreasing function φ: [0,+∞) → [0,+∞) such that φ(u) = o(u ln ln u) as u → ∞, and a modulus of continuity satisfying the condition (ln k)−1 = O(ω(nk−1)), we present an example of a function F ∈ φ(L) ∩ H1ω with an almost everywhere divergent subsequence {Snk(F, x)} of the sequence of partial sums of the trigonometric Fourier series of the function F.

We consider the multiplicative (in the sense of Vershik) probability measure corresponding to an arbitrary real dimension d on the set of all collections {Nj} of integer nonnegative numbers Nj, j = l0, l0 + 1, ..., satisfying the conditions $$ \sum\limits_{j = l_0 }^\infty {jN_j \leqslant M,} \sum\limits_{j = l_0 }^\infty {N_j = N} $$ , where l0, M, N are natural numbers. If M, N → ∞ and the rates of growth of these parameters satisfy a certain relation depending on d, and l0 depends on them in a special way (for d ≥ 2 we can take l0 = 1), then, in the limit, the “majority” of collections (with respect to the measure indicated above) concentrates near the limit distribution described by the Bose-Einstein formulas. We study the probabilities of the deviations of the sums Σj=l∞Nj from the corresponding cumulative integrals for the limit distribution. In an earlier paper (see [6]), we studied the case d = 3.

We consider functions f that are univalent in a plane angular domain of angle απ, 0 < α ≤ 2. It is proved that there exists a natural number k depending only on α such that the kth derivatives f(k) of these functions cannot be univalent in this angle. We find the least of the possible values of for k. As a consequence, we obtain an answer to the question posed by Kir’yatskii: if f is univalent in the half-plane, then its fourth derivative cannot be univalent in this half-plane.

Each additive cut in the nonstandard natural numbers *ℕ induces the equivalence relation MU on *ℕ defined as xMU y if |x − y| e U. Such equivalence relations are said to be monadic. Reducibility between monadic equivalence relations is studied. The main result (Theorem 3.1) is that reducibility can be defined in terms of cofinality (or coinitiality) and a special parameter of a cut, called its width. Smoothness and the existence of transversals are also considered. The results obtained are similar to theorems of modern descriptive set theory on the reducibility of Borel equivalence relations.

Earlier, Belegradek, Stolboushkin, and Taitslin proved that the collapse result holds in the theory of natural numbers with addition, i.e., each locally generic query using addition can be written without it. In this paper, we use the sufficient conditions of the collapse result obtained by Taitslin to prove that it holds in any extensions of the Presburger arithmetic by a unary function compatible with addition. The notion of a function compatible with addition was proposed by A. L. Semenov.

The asymptotics for the number of representations ofN asN→∞ is expressed as the sum of a number havingk prime divisors and a product of two natural numbers. The asymptotics is found fork≤(2−e) ln lnN and (2+e) ln lnN≤k≤b ln lnN, wheree>0. The results obtained are uniform with respect tok.

The smallest numberA

ON EXACT CONSTANTS IN THE WHITNEY THEOREM Yu. V. Kryakin 1. Introduction. For a function that is defined on the interval I = [0, 1], we put A~f(z) = E(-1) j f(z + jh). j=0 H. Whitney [1] proved the following result in 1957. THEOREM A. For any natural number n >_ 1 there is a positive number W(n) such that for a continuous function f(x) on I there exists an algebraic polynomial Pf of degree at most n -- 1 for which

A theorem is proved to the effect that if there exists a BIB-schema with parameters (pm−1,k, k−1), where k¦(pm−1), p is prime, and m is a natural number, then there exists a BIB-schema (pmn−1),k, k−1). A consequence is the existnece of a cyclic BIB-schema (pmn−1, pm−1, pm−2) (pm−1 is prime) that specifies each ordered pair of difference elements at any distance ρ = 1, 2, ..., pm−2 (cyclically) precisely once. Recursive theorems on the existence of difference matrices and (v, k, k)-difference families in the group Zv of residue classes mod v are proved, along with a theorem on difference families in an additive abelian group.

To each countable universal algebra there corresponds a class of natural numeration equivalences of numerations of this algebra. Conversely, to each equivalence ~ on the set of natural numbers m there corresponds a completely defined class of algebras that possess numerations with q as numeration equivalence. In this note we show that each algebra of an effective signature that has positive numeration with numeration equivalence of the form ~2 U id w, where ~ is a simple nonhypersimple subset of the set of natural numbers, has a continual lattice of congruences. We refer the reader to [1-3] for undefined notions. Everywhere in the sequel, the signature is assumed to be effective.

When a student first approaches the theory of infinite continued fractions a natural question that suggests itself is how to evaluate the expression

Let 1α + 2α + 3α + …. + nα be denoted by Sα. It is well known that Sα can be expressed as a polynomial in n of degree (α + 1). The expressions for S1, S2, S3 …. can be found in succession by elementary methods, which also give numerous relations such as S3 = 12S2S3 = 7S6 + 5S4. The elegant method which I am about to explain is not original. It is due in essence to the Arabian mathematician Alkarkhi (circa 1000 B.C.).